Mathematical modeling of plant cell fate transitions controlled by hormonal signals

Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to elucidate the impact of hormonal signaling on the fate transition rates between different zones corresponding to slowly dividing stem cells and fast dividing transit amplifying cells. The model is based on a simplified two-dimensional disc geometry of the SAM and accounts for a continuous displacement towards the periphery of cells produced in the central zone. Coupling growth and hormonal signaling results in a nonlinear system of reaction-diffusion equations on a growing domain with the growth rate depending on the model components. The model is tested by simulating perturbations in the level of key transcription factors that maintain SAM homeostasis. The model provides new insights on how the transcription factor HECATE is integrated in the regulatory network that governs stem cell differentiation.


Introduction
Tissue function is an effect of the cooperation of multiple specialized cell types. To establish, maintain and regenerate tissues, cell production and fate specification have to be orchestrated in a robust and well-defined manner. Perturbations of the underlying control mechanisms may reduce the ability of the organism to adapt to changing environmental conditions. Plants continuously generate new organs such as leaves, roots and flowers. For this purpose they maintain pools of stem cells which remain active during the whole life of the plant. The plant stem cells are located in specialized tissues, referred to as meristems. The accessibility of meristems to live-imaging and the repetitive formation of identical organs, such as leaves, make plants an attractive system to study the regulatory cues underlying cell production and fate transition.
Stem cell proliferation and fate choice have a direct impact on the architecture of the plant and its reproductive fitness. These vital functions require meristems to be robust with respect to perturbations such as injuries or environmental fluctuations. In case of agricultural plants, meristem dynamics are linked to the crop yield, therefore, a better understanding meristem regulations is of practical importance [26,27].
In this paper we focus on the stem cell dynamics in the shoot apical meristem (SAM) that is responsible for formation of all above ground structures. It has approximately a geometry of a two-dimensional disc. Stem cells are located in the central zone (CZ) surrounded by transit amplifying cells in the peripheral zone (PZ). Newly formed but still immature organs, so-called primordia, separate from the meristem at the outer boundary of the PZ.
A specialized cell population, the so called organizing center (OC), is located below the CZ and produces signals required for maintenance of the stem cell fate, see [10]. The location of the SAM and its morphology is summarized in Fig. 1.
There exists experimental evidence of different transcriptional factors coordinating cell proliferation and fate choice in the SAM [33]. A key regulatory loop of the SAM consists of the transcription factors WUSCHEL (WUS) and CLAVATA3 (CLV3). WUS is produced in the OC and moves to the CZ where it maintains the stem cell identity. Stem cells in turn produce CLV3 which inhibits WUS expression in OC cells [5,32]. This core feedback loop interacts with various other signals that fine-tune cell activity and allow the system to optimally adapt to environmental conditions [10]. One important example for such signals are the HECATE (HEC) transcription factors. It has been recently shown that HEC acts on cell fate transitions between the different SAM domains, however the underlying regulatory network remains elusive [11]. One possibility to understand the effects of perturbed signaling is the study of mutant phenotypes, such as the HEC mutant (hec123). However, studying the function of HEC genes at high spatio-temporal resolution is experimentally challenging [11].
To close this gap, we propose an integrated approach combining mathematical modeling with experimental manipulation and live imaging of plant stem cells.
Mathematical models are a powerful tool for studying complex nonlinear dynamics coordinated by multiple factors. They have contributed considerably to the understanding of SAM regulation [6,9,12,13,18,19,36,38]. In plants, cell fate decisions depend on local concentrations of spatially heterogeneous signals [10] and therefore, spatial models are required to describe meristem dynamics. For this purpose different approaches have been developed.
Individual-based models allow tracking dynamics of each individual cell that is explicitly modelled. Such approach has been applied e.g., to study patterning of the WUS expressing domain [6,18], mechanical signals [14], mechanisms of organ initiation [19], and cell fate determination [35,36]. On the other hand, a continuous approach based on reaction-diffusion equations and ordinary differential equation models allows to study spatio-temporal interactions of different signaling molecules. Such models have been applied e.g., to investigate cytokinin signaling [13] and patterning of the shoot apical meristem [9,38]. To investigate the impact of HEC on fate transition and proliferation rates of cells in the CZ and PZ, we have recently proposed a model based on the population dynamics approach, in which dynamics of different cell subpopulations are described by ordinary differential equations [11].
Such approach allows tracking how changes in cell proliferation, fate transition, primordia formation and primordia separation affect the time evolution and steady-state size of the different SAM zones but does not take into account spatio-temporal dynamics of the underlying signaling network.
In this paper, we study how time evolution of meristem cell populations and newly formed organs depends on the spatio-temporal dynamics of the underlying signaling network regulating cell self-renewal and differentiation. We develop a novel modeling framework that describes the SAM as a two-dimensional growing disc. The two-dimensional approximation of the domain is justified due to the SAM structure consisting of a small number of cell layers.
The model describes interactions of the plant meristem key signals (CLV3, WUS, Cytokinin and HEC) and links their local concentrations to cell proliferation and fate transition rates.
The change in the total SAM cell number is, in turn, linked to the change of domain size.
Coupling growth and signaling processes results in a nonlinear system of reaction-diffusion equations on a growing domain with the growth velocity depending on the model components.
Solving such problems is mathematically challenging. We implemented the model using the DUNE software package, which is a suitable numerical environment for sharp interface problems appearing in models with a growing domain [2][3][4].
The model was tested using recent experimental observations. Importantly, it allowed to gain more insight into stem cell differentiation dynamics in the HEC loss-of-function phenotype which has remained experimentally not feasible. A new insight stemming from this work is that HEC may reduce the differentiation rate of WUS producing cells [11].
The dynamics of OC cells are so far not well understood, since they are located deeply in the meristem and it is difficult to image them in vivo. Our model helps to understand how signals modulating the classical WUS-CLV3 loop act on the OC cells. In summary, the proposed comprehensive modeling and computational framework can be further used to generate hypotheses about interaction of the respective signaling factors and their impact on cell proliferation and fate transition. Plant cells are immobile since they are encased in a cell wall. Their fate is determined by local signals; for review see [10]. The SAM consists of multiple layers. Since the cell division process is anticlinal, i.e. the progeny of cells always belong to the same layer as their parent cells [24,28,31], we model only the uppermost cell layer, referred to as L1, together with the organizing center (OC). Specific cross-talk signals between the L1 and L2 layer have so far not been described. Cells in both layers are exposed to the WUS signal from the OC and regulate WUS expression by production of CLV3. We model the SAM as a two-dimensional disc with radius R. The OC is located below the center of the meristem and it is also disc-shaped. WUS is produced in the OC and transported to the CZ [7,35]. We model this process by a disc-shaped WUS source of radius r in the center of the meristem; see Hecate (HEC). The model accounts for the following processes: • WUS is produced by cells in the organizing center [7,35]. For simplicity we assume that WUS is produced at a constant rate per OC cell.
• We assume that CLV3 regulates the number of WUS-producing cells by increasing their differentiation rate or by decreasing their proliferation rate. This corresponds to a negative feedback loop between WUS and CLV3 [5,32].
However, HEC loss of function, as in the hec123 mutant, does not completely abrogate CK production. On the other hand, WUS loss of function leads to arrest of the meristem, i.e. loss of the stem cell population [20]. We therefore assume that CK production decreases to zero in absence of WUS and that CK production is maintained at low levels in absence of HEC.
In addition, we assume that all signals undergo a degradation at constant rates.
Description of the growing domain: In a good agreement with experimental data [11], we assume that all meristem cells have the same size. The radius of the meristem R(t) at time t can then be calculated based on the cell number. Let N (t) be the cell number at time t and α, β their proliferation and differentiation rates. Evolution of the cell population is governed by the equation Hence, for R(t) = N (t)/π, it holds d dt R(t) = (α(t) − β(t)) R(t) 2 .
Signal-dependent cell kinetics: We model the organizing center as a homogeneous cell population with signal-dependent proliferation or differentiation rate. We assume that CLV3 reduces proliferation or increases differentiation of OC cells and thus reduces the WUS concentrations. Similarly, HEC and CK reduce OC cell differentiation or induce proliferation and lead to increased WUS concentrations [13]. The considered regulatory network is summarized in Fig. 2.
We consider the following processes to describe evolution of the L1 SAM layer. WUS induces the stem cell fate [5,32]. Stem cells have lower division rates than transit amplifying cells [11,29]. An increase of WUS concentration leads to increase of the fraction of slowly-dividing stem cells in the meristem and decrease of cell production per unit of time.
Therefore, we assume that cell production decreases with increasing total WUS concentration. As shown by experiments, reduced CK activity leads to reduction of the meristem radius [16,34]. For this reason the growth rate of the meristem radius can become negative in presence of small CK concentrations. The mechanism underlying formation of organ primordia suggests that organ formation rates increase with the area of the meristem [8,15,19].
This implies that the cell outflux due to differentiation increases with increased meristem cell count, and hence it depends on the size of the domain and is proportional to R 2 . In accordance with the biological observations described above, we assume that high concentrations of CK and HEC lead to increased OC cell numbers and that high CLV3 concentrations lead to decreased OC cell numbers.
; it corresponds to a disc of radius R(t). The organizing center at time t is a discshaped domain of radius r(t) and it is denoted as Ω small (t). The centers of Ω(t) and Ω small (t) coincide. The diffusion constants of WUS, CLV3, CK and HEC are denoted by D i > 0, i = 1, . . . , 4, respectively. The above-listed assumptions result in the following system of equations: (1) with model parameters k i , d i , pp, R 0 , pdbasic being positive constants. Moreover, by χ Ω small (t) we denote the piecewise linear approximation of the indicator function of Ω small (t). The system is subjected to homogeneous Robin boundary conditions in u i and initial conditions for Model of the domain growth: The functions f and g are defined as follows: It is known that WUS expression decreases with increasing CLV3 concentrations [25]. We hypothesize that this is caused by a change of OC cell numbers. Based on experimental observations [11,13,21] we furthermore hypothesize that HEC and CK impact on OC dynamics.

This agrees with experiments showing that induction of HEC in the CZ results in an increase
of the OC [11]. In accordance with experimental observations showing repression of WUS by increased CLV3 concentrations [25], we consider CLV3 as the main regulator, in the sense that for high CLV3 concentrations the OC cell number decreases. We express the OC proliferation rate as the product of two functions, a decreasing Hill function depending on CLV3 and a function f that depends on CK and HEC. The shape of function f is depicted in Fig.   8 (A). Function f models our hypothesis that the organizing center grows if CK and HEC concentrations increase [11]. HEC is fine-tuning the meristem signaling. We assume that for increasing HEC concentrations the effect of CK on the meristem increases. This assumption follows the observations in [11]. For high HEC concentrations the CK effect saturates at a higher level compared to the case of low HEC concentrations (i.e., the impact of high levels of CK signaling is amplified by HEC). This may be explained by a HEC-induced production of CK target molecules. Since it has been observed experimentally that HEC and CK loss of function do not lead to loss of the OC [11,16], the value of f is positive for u 2 = u 3 = 0.
Experiments have shown that decreased concentrations of CK lead to smaller meris-tems [16,34]. This is modeled by the function g which is depicted in Fig. 8 (B) and reflects the observation that the meristem structure is robust to perturbations. Only large deviations of CK from its wild type concentration (either towards very high or very low concentrations) impact on the meristem radius by reducing the cell number [16].
It is not clear how regulation of OC size is accomplished. In principle, two extreme possibilities exist, namely constant proliferation and regulated differentiation or regulated proliferation and constant differentiation. The ODE for r as it is written above implies the latter. However, for uniformly bounded u i it can be rewritten as which corresponds to a constant proliferation K and a CLV3, HEC and CK dependent differentiation term. Here K denotes the maximum of k 77 1+k 7 u 1 f (u 2 , u 3 ). Taking into account the experimental results for the upper meristem layers [11], the first option, i.e. HECdependent regulation of differentiation seems more plausible.

Numerical approach
For numerical computations, the coupled system of diffusion-reaction equations (1) and domain evolvement (2) is decoupled using explicit equation splitting. The PDEs are then solved by the moving finite element method [23] using conforming piecewise bilinear finite elements on quadrilaterals in space and the implicit Euler method in time. The arising nonlinear algebraic system is solved with Newton's method where the (linear) Jacobian system is solved with a sparse direct solver. The ODEs for domain movement are discretized by the explicit Euler method. Implementation has been carried out in the PDE software framework Dune/PDELab [2][3][4].

Model calibration
Initial data and model parameters: Since stem cells are identified experimentally using CLV3 reporters, we define them in the model as the cells located at positions where CLV3 concentration is above a certain threshold. We assume that the meristem of the unperturbed adult wild type plant is in a steady-state. Experiments show that under such conditions the CZ cell number corresponds to approximately 10% of the total meristem cell number [11]. This ratio is hold in the locally stable equilibrium depicted in Fig. 6 Table 1: Parameter values corresponding to the wild type (unperturbed) scenario.
Stationary state: The stationary state which serves as departure point for the simulated experiments has been found numerically. The initial condition used to converge to this equilibrium is depicted in Fig. 3 (A), the time evolution of R and r in Fig. 3 (B), (C). In addition, we provide mathematical evidence for the local stability of this steady state.  Figure 4: The blue curve describes the zero level-set of function G. The points (R 1 , 0) and (R 2 , R 2 ) correspond to the intersection of the zero level-set of function G with the lines r = 0 and r = R. The value of the function F in these points is negative. For r = 0 and r = R we are able to solve (1) explicitly. Moreover, we know that there exists at least one point (R,r) for which function F is positive. The latter is a consequence of the parameter choice.
Thus there exists at least one point (R * , r * ) such that: G(R * , r * ) = 0, F (R * , r * ) = 0 and in the neighborhood of this point for R > R * it holds F (R, r * ) ≥ 0 and for R < R * it holds F (R, r * ) < 0. Further on, we will consider the stability of the steady state solution (R * , r * ).
Assuming that the reaction-diffusion process of signaling molecules is faster than changes of the domain radius, we obtain a quasi-stationary system. For a given pair (R, r) we solve the quasi-stationary problem for u, i.e. setting the time derivatives in (1) where functionsF (u, R, r) andG(u, R, r) correspond to the expressions on the right-hand of system (2). Linearization in the neighborhood of the steady-state (R * , r * ) leads to since F (R * , r * ) = 0 and G(R * , r * ) = 0. Derivatives of G are negative, what can be checked by explicit calculations. The main task is to calculate derivatives of function F . We are not able to do it analytically. However, using our numerical approach we can calculate values of the function F in the neighborhood of the steady-state, see Fig. 5. Hence, we obtain that ∂ r F is negative and ∂ R F is positive.
since r * and R * are positive. This implies that the steady-state is locally stable.
Experiments used to test the model: The model is tested by comparing its results to the following experiments.
• WUS over-expression in the whole meristem: This leads to radial expansion of the CLV3 expressing domain. The change of total SAM size is negligible [37].
• WUS loss of function: This leads to termination of the meristem and loss of stem cells [20].
• CLV3 over-expression in the central zone: This leads to repression of WUS, it has been shown that a 10 fold change in CLV3-expression levels does not affect meristem size [25].
• CLV3 loss of function: This leads to larger meristems and expansion of the CZ [30].
• Reduced degradation of CK: This leads to larger meristems and a larger OC [1].
• HEC over-expression by stem cells: This leads to expansion of the central zone and subsequent loss of meristem structure [11].
• HEC loss of function: The HEC triple mutant hec123 expresses no functional HEC.
This leads to significantly smaller meristems compared to the wild type [11].

Simulation of key experiments
In this section, we test the model comparing it to the outcomes of a set of experiments involving over-expression or loss-of-function of certain signals. In several places, we refer to genes expressed under a promoter. To express gene X under the promoter of gene Y means to engineer genes such that X is always expressed together with Y. If the promoter of Y is ubiquitously expressed, then X is expressed everywhere in the meristem, if the promoter of Y is site specific, then X is expressed only in a subdomain of the meristem.

Perturbation of WUS
WUS over-expression: There exist different experimental works studying an ubiquituous increase of WUS [22,37] Experimentally this has been accomplished using a glucocorticoidinducible form of WUS under a promoter that causes ubiquitous expression. The experimental setting is modelled by the following modification of the equation for WUS: The positive constant c denotes the rate of WUS over-expression which is independent of space, time and other signals. The steady-state shown in Fig. 6 (A) serves as initial condition for simulation of the experiment.
The biological experiments agree in the observation that the central zone gets larger. This we also observe in the simulations. If the over-expression is high enough, the simulations show a radial expansion of the CLV3 expression domain and the system converges to a state where CLV3 is expressed in the whole meristem; Fig. 6 (B). This observation agrees with the experimental results from [37]. The radial expansion of the CLV3 expressing domain is depicted in Fig. 10. Biologically it has been considered unexpected that ubiquitous WUS over-expression leads to a radial growth of the CZ instead of a simultaneous up-regulation of CZ fate in the whole meristem. It has been speculated that the reason for this observation is that PZ cells located at the boundary of the CZ respond differently to WUS compared to other PZ cells [37]. The model simulations, however, suggest that the experimental observations can be explained even if all PZ cells respond to WUS equally. The observation that the total meristem size does not change significantly in the simulation matches also the experimental findings [37].
Simulations predict different outcomes for different levels of over-expression, Fig. 9 in Supplement. Since it is experimentally difficult to fine-tune the rate of over-expression, the diverse outcomes obtained in the simulations have not yet been observed.

WUS loss-of-function:
To simulate WUS loss of function, we set the WUS concentration equal to zero in the equations for CLV3, HEC and CK. In this setting WUS is still produced but it does not impact on dynamics of other signals. This corresponds e.g., to the case where WUS cannot bind to its receptors. This scenario results in loss of stem cells, since the defective WUS protein cannot induce production of CLV3. Experimentally such a scenario can be studied using WUS loss-of-function mutants, as it has been done in [20]. The loss of stem cells observed in the simulations is in agreement with experimental data. Due to the low CLV3 levels, there exists no feedback inhibition of WUS expression which implies that levels of the non-functional WUS protein are high and the organizing center grows. For this reason the system converges to a state with negligible CLV3 concentrations and high WUS expression as shown in Fig. 6 (C). Time evolution of R and r is depicted in Fig. 11.

Perturbation of CLV3
We simulate a scenario where CLV3 expression is increased in its natural expression domain.
This has been experimentally accomplished in [25] by using an ethanol-inducible CLV3 construct that is expressed under the CLV3 promoter. We implement this by multiplying the CLV3 production term by a positive constant c: Model simulations predict the outcome of experiments showing only mild changes in total meristem size. We also obtain reduced WUS concentrations. Such dynamics have been reported as a transient phenomenon in experiments [25]. Simulation results are shown in Fig. 6 (D) and Fig. 12.
We implement CLV3 loss-of-function by setting CLV3 concentrations to zero in the right hand-side of the equations for WUS, HEC, CK, r and R. Experimentally such a scenario has been accomplished through CLV3 silencing using inducible RNA interference [30]. As in experiments, we observe expansion of the central meristem zone [30]. However, we do not observe the reported expansion of the total SAM size. This suggests that there exists a coupling between CZ size and PZ proliferation rate that is not considered in the model and has not yet been characterized in detail. Results are shown in Fig. 6 (E) and Fig. 13.

Perturbation of CK
We simulate the following scenarios of CK perturbation: • CK over-expression: during this experiment we change the degradation rate in the equation for CK; • CK loss of function: during this experiment we put 0 instead of the production term in the CK-equation.
To study the impact of increased CK concentration the ckx3 ckx5 double mutant has been used. In this mutant the degradation of CK via CKXs (cytokinin oxigenases/dehydrogenases) is reduced [1]. We model this experiment by reducing the value of d 2 , which corresponds to decreased CK degradation, as in the experiments. Numerical simulations are consistent with the experiments in showing an increase of the OC, a slight increase of the CZ and an increase in meristem radius [1].
Arabidopsis mutants lacking functional CK receptors, such as the cre1-12 ahk2-2 ahk3-3 triple mutant allow to study CK loss of function [16]. In agreement with experiments [16,34] simulations show a decrease of the OC radius and of the total meristem size, see Figure 15. show a change of meristem size and stem cell number in case of HEC over-expression [11].
Our hypothesis is that HEC acts on the OC cell differentiation or proliferation. This cannot be directly monitored in experiments but can be tested using our model. We implement this experiment in the model by adding a HEC production term that is proportional to the CLV3 source term in equation (1). This yields the following equation for HEC: where c describes the proportionality between HEC and CLV3 production resulting from the expression of both signals under the same promoter. This scenario corresponds to the experimental setting from [11], where HEC is expressed in stem cells that are characterized by high CLV3 levels. For c large enough we observe that the system converges to a state with constant in space signal concentrations. This corresponds to the experimentally observed expansion of the CZ towards the boundaries of the meristem. Identically as in the experiments WUS, CLV3 and CK concentrations are increased compared to the wild-type meristem, Fig. 6 (H) for c = 3. In agreement with experiments we observe an increase in total meristem radius R; Fig. 16.
We implement HEC loss of function by setting HEC production equal to zero. As in experimental data [11], we observe mild changes in WUS and CLV (Fig. 6 (I)) that are associated with a smaller meristem (Fig. 17).

Model simulations suggest a direct effect of HEC on OC cells
In Fig. 2  Our simulations lead to several new biological insights. It is known that HEC is not expressed in OC cells and that experimental expression of HEC in the OC leads to the loss of the meristem [33]. Our model suggests that HEC directly acts on OC cell kinetics and that this action is required to observe the growth of the CZ in HEC over-expression experiments.
In addition to this there exists a CK mediated effect of HEC on OC cell differentiation: HEC increases CK expression and CK affects OC cells. This HEC mediated increase of CK signaling is required to explain the reduced CK levels observed in the hec1,2,3 triple mutant [11]. The effect of CK on OC cell differentiation explains the change of OC size under CK perturbation. Together these control couplings constitute an indirect CK-mediated effect of HEC on OC cell differentiation. It is difficult to predict intuitively whether this indirect effect is sufficient to explain experimental results or whether an additional direct action of HEC on the OC is required. Our simulations suggest that without the direct effect the WUS expressing domain does not show the gradual increase until it reaches the boundary of the meristem as it has been observed experimentally in [11]. Therefore, our simulation support the existence of a direct effect of HEC on OC cell kinetics. The investigation of HEC target genes may help to identify reguatory nodes mediating the effect of HEC and to integrate them with known candidates such as NGATHA [17].
In case of CLV3 loss of function and increased CK activity the model correctly predicts an increase of the CZ, however it fails to reproduce the observed growth of the meristem.
The reason for this is that in the model system PZ cells are recommitted to the stem cell fate which leads to an increase of slowly dividing stem cells at the expense of fast dividing PZ cells. The discrepancies with experimental data suggest that there exists a mechanism coupling the CZ and PZ dynamics that is not included in our model. Such a coupling has already been described in [30] where an increase of PZ cell proliferation rates in case of CZ expansion has been observed. Our simulations suggest that CK signaling is not sufficient to mediate this effect. In [37] it is discussed that a non-monotonous dependence of proliferation rates on WUS could play a role in this context.
Future version of the model have to include auxin signaling. It is known that auxin promotes cell fate transition from the PZ to lateral organs. Auxin signaling is repressed by WUS [22] in the meristem center and modulated by HEC in the meristem periphery [11]. The proposed computational framework is ideally suited to investigate details of these interactions.
In conclusion, we have developed a modeling framework that allows to study how gene

A Supplement
A.1 Functions f and g The shapes of the functions f and g are depicted in Fig. 8. A.2 Different rates of WUS over-expression We run simulations for different rates of WUS over-expression. In the model this corresponds to different values of c. Depending on the rate of WUS over-expression different dynamics can be observed.
• Mild WUS over-expression (c = 1.0) in the whole meristem, Fig. 9 (A): Inhibition of HEC leads to increased OC differentiation. The OC shrinks and therefore also the WUS and CLV3 concentration in the center of the meristem. The over-expression is too mild to trigger a CLV3 production comparable to that in the center of the wild type CZ. The system approaches a state with decreased WUS and CLV3 concentrations.
The OC cell number is different from zero and the state is nonconstant in space.
• Intermediate WUS over-expression (c = 2.9) in the whole meristem, Fig. 9 (B): Inhibition of HEC leads to increased OC differentiation and extinction of the OC cells.
Therefore, WUS production in the OC ceases. The system converges to a state that is constant in space and that is maintained by the ubiquitous constant WUS expression.
This level of WUS expression, however, leads to CLV3 concentrations lower than the CLV3 concentrations in the center of the wild-type meristem.
• Strong WUS over-expression (c = 3.1) in the whole meristem, Fig. 9    A.4 CLV3 over-expression  A.6 Reduced CK degradation ∂ t u 2 = 0.2∆u 2 + 1 + 4u 3 1 + 0.1u 3 Reduced CK degradation leads to an increase of the OC radius. If the reduction is strong enough, we also observe an increase in meristem radius.

A.8 HEC over-expression in stem cells
As in experiments HEC over-expression in stem cells leads to an increase of the meristem radius and the OC. The OC radius approaches the radius of the the total meristem, Fig. 16.
Different levels of over-expression result in qualitatively similar dynamics. If we rerun the simulations omitting the direct effect of HEC on OC cell differentiation, the OC radius does not approach the meristem radius, see Fig. 7 (C-D) which is in contradiction to experimental observations [11].
To simulate latter scenario, in the ODE for r the HEC concentration was fixed to 0.3.